# Questions tagged [sheaf-cohomology]

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### Sheaf cohomology definition

I have seen multiple definitions for sheaf cohomology and wanted to ask for the reason. One goes through injective resolutions and the other through flasque resolutions. For paracompact bases it holds ...
• 11
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### Why does the associated sheaf vanish?

I am learning local cohomology from Hartshorne’s book Local Cohomology. My question is about understanding a line in the proof of proposition 1.11 in this book. The set-up for proposition 1.11 is that ...
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1 vote
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### Sheaf cohomology in number theory

I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...
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### Using higher topos theory to study Cech cohomology

It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...
• 821
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### Singular cohomology to cohomology of quasi-coherent sheaf

Let $X$ be a projective nonsingular variety (integral Noetherian scheme of finite type that is proper over a field $k=\overline{k}$ such that $\Omega^1_X$ is locally free). Suppose one knows the ...
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### Unbounded acyclic resolutions

Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...
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### Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper

In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
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### Is $\mathbb{C}^*$ not irreducible, or is every locally constant sheaf on $\mathbb{C}^*$ constant?

I am running into contradiction from the following set of definitions, propositions, and assumptions. Can anyone spot where I'm off? Definition A sheaf $\mathcal{F}$ on a topological space $X$ is ...
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### Formula for the Euler characteristic of a local system on $\mathbb{P}^1$

Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion. Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...
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### Isomorphism of cohomology bundles for smooth homotopic fibrations

Let $M,N$ be closed smooth manifolds. Let $f_0,f_1\colon M\to N$ be two smooth fibrations which are homotopic to each other in the class of smooth (equivalently, continuous) maps. Let $E^i_0, E^i_1$ ...
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### Zorn's lemma for Grothendieck sites

In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction ...
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### Is the right adjoint to presheaf direct image interesting?

Let $X\overset{f}{\to}Y$ be a continuous map. It induces on presheaves a classical adjunction inverse image ⊣ direct image. However, the direct image functor has a further right adjoint, defined by ...
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### Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?

Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
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### On the upper-bound for a type of quintuple Kloosterman sums

Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound. My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...
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### Global section of pullback of an ideal sheaf

For a local ring $R$ with maximal ideal $\mathfrak{m}\subset R$ and residue field $\kappa$, and a flat morphism $f\colon X\rightarrow \mathrm{Spec} R$ of schemes, we consider the short exact sequence ...
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### The cohomology groups corresponding to a modified global sections functor

Let $\mathcal{F}$ be a sheaf on the big etale site of $Sm_k$. I am looking for a way to calculate a modified version of sheaf cohomology. Let $X$ be a smooth scheme and $Z$ a closed sub-scheme. After ...
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1 vote
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### Continuity of motivic cohomology under direct limit

Given the motivic complexes $\mathbb{Z}(n)$ on the big Zariski site of finite type smooth $k$-schemes denoted by $FinSm_k$, we pullback it to the smooth $k$-schemes i.e. $Sm_k$. For example for a ...
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1 vote
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### Künneth formula for local cohomology with support

In "Differential operators on the flag varieties" (http://www.numdam.org/article/AST_1981__87-88__43_0.pdf) by Brylinski, he uses on page 53 a Künneth formula for local cohomology with ...
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### Divisors whose restriction is big

Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$. ...
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### Higher direct image with compact support of a constant sheaf

Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A_X$ on $X$, the higher direct images $R^n f_* A_X$ are ...
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### Global sections of multiples of a divisor

Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$. Is there any example where $h^0(X,mD) = 0$ ...
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### Extending $G$-torsors on open subsets of affine space

Let $k$ be a characteristic zero field, $V \subset \mathbb{A}^n_k$ an open subscheme, $G$ a split reductive group over $k$ and $T$ a $G$-torsor over $V$ (in the etale, equivalently fppf topology). ...
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### Cohomology of Grothendieck topology

My naïve cartoon picture of the construction of étale cohomology is this: start with a scheme, associate to it a Grothendieck topology (making a site). A functor from the Grothendieck topology to ...
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### Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$
Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$. I ...